The greatest calamity in the history of science

The greatest calamity in the history of science was the failure of Archimedes to inventpositional notation. – Carl Friedrich Gauss

Although it has increased in the past 20 years, the number of mathematicians in the United States who devote large amounts of time to thinking about mathematics education below the college level is rather small. As a consequence, there is no standard set of activities that define the occupation. Teacher preparation and professional development are obvious areas for work, but at my university, we don’t have an education school, and no courses specifically for training teachers. So, I have had to figure out for myself what I might do to improve mathematics education (besides serving on committees and attending workshops, of course!).

What I have mainly done is, try to think about the mathematics of elementary school: what it involves, and what is needed to develop it robustly – the kinds of ideas that might go into writing a set of standards for mathematics education. I have found this to be a serious intellectual challenge. It is this challenge that has kept me thinking about mathematics education. After reading the book Knowing and Teaching Elementary Mathematics by Liping Ma, and serving on the committee that produced Adding It Up, I had the sense that place value was a topic that needed more attention in US math instruction. As part of a series of workshops at the Park City Mathematics Institute, whose goal was to have mathematicians attend to and discuss aspects of the elementary curriculum, I undertook to write an essay about place value. Accordingly, I sacrificed a substantial part of a visit to the wonderful Newton Institute in Cambridge, England, to produce Taking Place Value Seriously (joint with Susanna Epp; available online at http://www.maa.org/pmet/resources/PVHoweEpp-Nov2008.pdf). (Not to complain too much – I also wrote a research paper.)

One of biggest surprises of that project was the realization that there was no standard short name for the basic building blocks of the decimal system, the numbers like 2 and 70 and 300 and 4,000, that have only one non-zero digit. Such a number can be described as a “digit times a power of 10”, but this is rather cumbersome; also, it assumes the concept of “power of a number”, while one would like to draw student attention to these numbers before that concept is available. So I gave them a name. In the first draft of the essay, I called them “very round number”s. However, some of my Park City colleagues found this insufficiently dignified, so the term that survived to the essay is “single place number”. After writing the essay, I took every opportunity I had to advocate more attention to place value in US math education. (I found that Singapore does a much better job with place value, which reassured me that I was not crazy to emphasize it.) But I gradually realized that I was not communicating what I wanted to, and that the reason was connected to the lack of a name for a “digit times a power of 10″. Place value in the US is treated as a vocabulary issue: ones place, tens place, hundreds place. It is described procedurally rather than conceptually. Place value is treated as the rule that tells you how to read a number, not as the idea of expressing any number as a sum numbers of a very special form:”digit times a power of 10”.

It is this idea that gives place value notation its amazing power. Not only can we write very large numbers efficiently, we can compute with them by simple algorithms, based on two fundamental principles: 1) know how compute the sum or product of two “digit times a power of 10″s (in addition, only worry about the case when the powers of ten are the same); and 2) combine the results of 1) using the Rules of Arithmetic.

Because of the particular nature of the special numbers (they are “digit times a power of 10″s!), step 1) largely boils down to knowing how to compute with digits: the addition facts and the multiplication facts. This small amount of memorization gives one the potential to do an infinite number of computations.

But talking about place value procedurally prevents us from saying this. It takes the focus away from the nature of the special numbers and what we are doing with them, and turns it to the question of how to read numbers. This is of course very important, but it is only the very beginning, the first step in understanding the marvelous conceptual leap that place value embodies.

So my plea to the mathematics education community is: let’s talk conceptually about place value, and help teachers think conceptually about place value. And as an enabling first step, let’s find a short catchy standard name for “digit times a power of 10”!

Responses

@Anonymous: I love the term “monomial”, perhaps for a wrong reason: I have to teach long division of polynomials to Foundation Studies (that is, Year 0) undergraduates (mostly from various programmes in engineering), and have to refer, of course, to long division of integers (which many of my students have only a very shaky idea about).

Interesting. In the OEIS, these numbers are the 37124th sequence: http://oeis.org/A037124 I’m astonished that it took 37 thousand entries before something this fundamental was added, and this speaks to the unawareness even professional mathematicians have for these numbers. BTW, the OEIS calls them “Numbers that contain only one nonzero digit.”

Personally, I LOVE the term “very round numbers.” Maybe I spend too much of my time with kids? And for teaching, I think the place value cards for expanding numbers are very useful — they help students see how you can break the number apart and work with each piece individually, and then put your answer back together in the end.